Lassa fever is a fatal zoonotic hemorrhagic disease caused by Lassa virus carried by multimammate rats, which are widely spread in West Africa. In this work, a fractional-order model for Lassa fever transmission dynamics is developed and analysed. The model involves transmissions from rodents-to-human, person-to-person, as well as from Lassa virus infested environment/surfaces. The basic properties of the model such as positivity of solutions, and local stability of the disease-free equilibrium are determined. The reproduction number, ${\mathcal{R}}_0$, of the model is determined using the next generation method and it is used to determine the suitable conditions for disease progression as well as its containment. In addition, we performed sensitivity analysis of the model parameters using the Latin Hypercube Sampling (LHS) scheme to determine the most influential processes on the disease threshold, and determined the key processes to be focused on if the infection is to be curtailed. Moreover, fixed point theory was used to prove the existence and uniqueness of non-trivial solutions of the model. We used the Adams-Bashforth Moulton method to solve the model system numerically for different orders of the fractional derivative. Our results show that using various interventions and control measures such as controlling environmental contamination, reducing rodents-to-humans transmission and interpersonal contact, can significantly help in curbing new infections. Morestill, we observe that an increase in the memory effect, i.e. dependence on future values of the model on the previous states predicts lower peak values of infection cases in the short term, but higher equilibrium values in the long term.